First Course in the Theory of Equations, by Leonard Eugene Dickson pptx

The Project Gutenberg EBook of First Course in the Theory of Equations, by Leonard Eugene Dickson This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: First Course in the Theory of Equations Author: Leonard Eugene Dickson Release Date: August 25, 2009 [EBook #29785] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK THEORY OF EQUATIONS *** Produced by Peter Vachuska, Andrew D. Hwang, Dave Morgan, and the Online Distributed Proofreading Team at http://www.pgdp.net Transcriber’s Note This PDF file is formatted for printing, but may be easily formatted for screen viewing. Please see the preamble of the L A T E X source file for instructions. Table of contents entries and running heads have been normalized. Archaic spellings (constructible, parallelopiped) and variants (coordinates/coördinates, two-rowed/2-rowed, etc.) have been retained from the original. Minor typographical corrections, and minor changes to the presentational style, have been made without comment. Figures may have been relocated slightly with respect to the surrounding text. FIRST COURSE IN THE THEORY OF EQUATIONS BY LEONARD EUGENE DICKSON, Ph.D. CORRESPONDANT DE L’INSTITUT DE FRANCE PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Limited Copyright, 1922, by LEONARD EUGENE DICKSON All Rights Reserved This book or any part thereof must not be reproduced in any form without the written permission of the publisher. Printed in U. S. A. PRESS OF BRAUNWORTH & CO., INC. BOOK MANUFACTURERS BROOKLYN, NEW YORK PREFACE The theory of equations is not only a necessity in the subsequent mathe- matical courses and their applications, but furnishes an illuminating sequel to geometry, algebra and analytic geometry. Moreover, it develops anew and in greater detail various fundamental ideas of calculus for the simple, but impor- tant, case of polynomials. The theory of equations therefore affords a useful supplement to differential calculus whether taken subsequently or simultane- ously. It was to meet the numerous needs of the student in regard to his earlier and future mathematical courses that the present book was planned with great care and after wide consultation. It differs essentially from the author’s Elementary Theory of Equations, both in regard to omissions and additions, and since it is addressed to younger students and may be used parallel with a course in differential calculus. Simpler and more detailed proofs are now employed. The exercises are simpler, more numerous, of greater variety, and involve more practical applications. This book throws important light on various elementary topics. For ex- ample, an alert student of geometry who has learned how to bisect any angle is apt to ask if every angle can be trisected with ruler and compasses and if not, why not. After learning how to construct regular polygons of 3, 4, 5, 6, 8 and 10 sides, he will be inquisitive about the missing ones of 7 and 9 sides. The teacher will be in a comfortable position if he knows the facts and what is involved in the simplest discussion to date of these questions, as given in Chapter III. Other chapters throw needed light on various topics of algebra. In particular, the theory of graphs is presented in Chapter V in a more scientific and practical manner than was possible in algebra and analytic geometry. There is developed a method of computing a real root of an equation with minimum labor and with certainty as to the accuracy of all the decimals ob- tained. We first find by Horner’s method successive transformed equations whose number is half of the desired number of significant figures of the root. The final equation is reduced to a linear equation by applying to the con- stant term the correction computed from the omitted terms of the second and iv PREFACE higher degrees, and the work is completed by abridged division. The method combines speed with control of accuracy. Newton’s method, which is presented from both the graphical and the numerical standpoints, has the advantage of being applicable also to equations which are not algebraic; it is applied in detail to various such equations. In order to locate or isolate the real roots of an equation we may employ a graph, provided it be constructed scientifically, or the theorems of Descartes, Sturm, and Budan, which are usually neither stated, nor proved, correctly. The long chapter on determinants is independent of the earlier chapters. The theory of a general system of linear equations is here presented also from the standpoint of matrices. For valuable suggestions made after reading the preliminary manuscript of this book, the author is greatly indebted to Professor Bussey of the University of Minnesota, Professor Roever of Washington University, Professor Kempner of the University of Illinois, and Professor Young of the University of Chicago. The revised manuscript was much improved after it was read critically by Professor Curtiss of Northwestern University. The author’s thanks are due also to Professor Dresden of the University of Wisconsin for various useful suggestions on the proof-sheets. Chicago, 1921. CONTENTS Numbers refer to pages. CHAPTER I Complex Numbers Square Roots, 1. Complex Numbers, 1. Cube Roots of Unity, 3. Geometrical Representation, 3. Product, 4. Quotient, 5. De Moivre’s Theorem, 5. Cube Roots, 6. Roots of Complex Numbers, 7. Roots of Unity, 8. Primitive Roots of Unity, 9. CHAPTER II Theorems on Roots of Equations Quadratic Equation, 13. Polynomial, 14. Remainder Theorem, 14. Synthetic Division, 16. Factored Form of a Polynomial, 18. Multiple Roots, 18. Identical Polynomials, 19. Fundamental Theorem of Algebra, 20. Relations between Roots and Coefficients, 20. Imaginary Roots occur in Pairs, 22. Upper Limit to the Real Roots, 23. Another Upper Limit to the Roots, 24. Integral Roots, 27. Newton’s Method for Integral Roots, 28. Another Method for Integral Roots, 30. Rational Roots, 31. CHAPTER III Constructions with Ruler and Compasses Impossible Constructions, 33. Graphical Solution of a Quadratic Equation, 33. Analytic Criterion for Constructibility, 34. Cubic Equations with a Constructible Root, 36. Trisection of an Angle, 38. Duplication of a Cube, 39. Regular Polygon of 7 Sides, 39. Regular Polygon of 7 Sides and Roots of Unity, 40. Reciprocal Equations, 41. Regular Polygon of 9 Sides, 43. The Periods of Roots of Unity, 44. Regular Polygon of 17 Sides, 45. Construction of a Regular Polygon of 17 Sides, 47. Regular Polygon of n Sides, 48. v vi CONTENTS CHAPTER IV Cubic and Quartic Equations Reduced Cubic Equation, 51. Algebraic Solution of a Cubic, 51. Discrimi- nant, 53. Number of Real Roots of a Cubic, 54. Irreducible Case, 54. Trigono- metric Solution of a Cubic, 55. Ferrari’s Solution of the Quartic Equation, 56. Resolvent Cubic, 57. Discriminant, 58. Descartes’ Solution of the Quartic Equa- tion, 59. Symmetrical Form of Descartes’ Solution, 60. CHAPTER V The Graph of an Equation Use of Graphs, 63. Caution in Plotting, 64. Bend Points, 64. Derivatives, 66. Horizontal Tangents, 68. Multiple Roots, 68. Ordinary and Inflexion Tangents, 70. Real Roots of a Cubic Equation, 73. Continuity, 74. Continuity of Polynomials, 75. Condition for a Root Between a and b, 75. Sign of a Polynomial at Infinity, 77. Rolle’s Theorem, 77. CHAPTER VI Isolation of Real Roots Purpose and Methods of Isolating the Real Roots, 81. Descartes’ Rule of Signs, 81. Sturm’s Method, 85. Sturm’s Theorem, 86. Simplifications of Sturm’s Functions, 88. Sturm’s Functions for a Quartic Equation, 90. Sturm’s Theorem for Multiple Roots, 92. Budan’s Theorem, 93. CHAPTER VII Solution of Numerical Equations Horner’s Method, 97. Newton’s Method, 102. Algebraic and Graphical Dis- cussion, 103. Systematic Computation, 106. For Functions not Polynomials, 108. Imaginary Roots, 110. CHAPTER VIII Determinants; Systems of Linear Equations Solution of 2 Linear Equations by Determinants, 115. Solution of 3 Linear Equa- tions by Determinants, 116. Signs of the Terms of a Determinant, 117. Even and Odd Arrangements, 118. Definition of a Determinant of Order n, 119. Interchange of Rows and Columns, 120. Interchange of Two Columns, 121. Interchange of Two Rows, 122. Two Rows or Two Columns Alike, 122. Minors, 123. Expansion, 123. Removal of Factors, 125. Sum of Determinants, 126. Addition of Columns or Rows, 127. System of n Linear Equations in n Unknowns, 128. Rank, 130. Sys- tem of n Linear Equations in n Unknowns, 130. Homogeneous Equations, 134. System of m Linear Equations in n Unknowns, 135. Complementary Minors, 137. CONTENTS vii Laplace’s Development by Columns, 137. Laplace’s Development by Rows, 138. Product of Determinants, 139. CHAPTER IX Symmetric Functions Sigma Functions, Elementary Symmetric Functions, 143. Fundamental Theo- rem, 144. Functions Symmetric in all but One Root, 147. Sums of Like Powers of the Roots, 150. Waring’s Formula, 152. Computation of Sigma Functions, 156. Computation of Symmetric Functions, 157. CHAPTER X Elimination, Resultants And Discriminants Elimination, 159. Resultant of Two Polynomials, 159. Sylvester’s Method of Elimination, 161. Bézout’s Method of Elimination, 164. General Theorem on Elimination, 166. Discriminants, 167. APPENDIX Fundamental Theorem of Algebra Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 [...]... divide the terms of our equation (7) by c0 , the essential part of our theorem is contained in the following simpler statement: Corollary In an equation in x of degree n, in which the coefficient n is unity, the sum of the n roots is equal to the negative of the coefficient of x of xn−1 , the sum of the products of the roots two at a time is equal to the coefficient of xn−2 , etc.; finally the product of all the. .. i sin(θ + α) 6 Quotient of Complex Numbers Taking α = β − θ in (2) and dividing the members of the resulting equation by cos θ + i sin θ, we get cos β + i sin β = cos(β − θ) + i sin(β − θ) cos θ + i sin θ Hence the amplitude of the quotient of R(cos β+i sin β) by r(cos θ+i sin θ) is equal to the difference β − θ of their amplitudes, while the modulus of the quotient is equal to the quotient R/r of their... Ex 2 the points representing the six sixth roots of unity Obtain this result another way 4 Find the five fifth roots of −1 5 Obtain the trigonometric forms of the nine ninth roots of unity Which of them are cube roots of unity? 6 Which powers of a ninth root (7) of unity are cube roots of unity? 11 Primitive nth Roots of Unity An nth root of unity is called primitive if n is the smallest positive integral... value of k, (5) is an answer since its nth power reduces to cos A + i sin A by DeMoivre’s theorem Next, the value n of k gives the same answer as the value 0 of k; the value n + 1 of k gives the same answer as the value 1 of k; and in general the value n + m of k gives the same answer as the value m of k Hence we may restrict attention to the values 0, 1, , n − 1 of k Finally, the answers (5) given by. .. Product of Complex Numbers By actual multiplication, r(cos θ + i sin θ) r (cos α + i sin α) = rr (cos θ cos α − sin θ sin α) + i(sin θ cos α + cos θ sin α) = rr cos(θ + α) + i sin(θ + α)], by trigonometry Hence the modulus of the product of two complex numbers is equal to the product of their moduli, while the amplitude of the product is equal to the sum of their amplitudes For example, the square of ω... sin n n By De Moivre’s theorem, the general number (6) is equal to the kth power of R Hence the n distinct nth roots of unity are (8) R, R2 , R3 , , Rn−1 , Rn = 1 As a special case of the final remark in §9, the n complex numbers (6), and therefore the numbers (8), are represented geometrically by the vertices of a regular polygon of n sides inscribed in the circle of radius unity and center at the. .. quotient of two complex numbers 9 nth Roots As illustrated in §8, it is evident that the nth roots of any complex number ρ(cos A + i sin A) are the products of the nth roots of cos A + i sin A by the positive real nth root of the positive real number ρ (which may be found by logarithms) Let an nth root of cos A + i sin A be of the form (4) r(cos θ + i sin θ) Then, by De Moivre’s theorem, rn (cos nθ + i sin... line, ignoring the powers of x 1 3 2 0 10 −2 20 −5 36 1 5 10 18 31 2 First we bring down the first coefficient 1 Then we multiply it by the given value 2 and enter the product 2 directly under the second coefficient 3, add and write the sum 5 below Similarly, we enter the product of 5 by 2 under the third coefficient 0, add and write the sum 10 below; etc The final number 31 in the third line is the value of. .. are the remaining two roots of the given equation, and find these roots 6 If x4 − 2x3 − 12x2 + 10x + 3 = 0 has the roots 1 and −3, find the remaining two roots 7 Find the quotient of 2x4 − x3 − 6x2 + 4x − 8 by x2 − 4 8 Find the quotient of x4 − 3x3 + 3x2 − 3x + 2 by x2 − 3x + 2 9 Solve Exercises 1, 2, 3, 6, 7 of §14 by synthetic division 18 THEOREMS ON ROOTS OF EQUATIONS [Ch II 16 Factored Form of a... COEFFICIENTS 21 These results may be expressed in the following words: Theorem If α1 , , αn are the roots of equation (7), the sum of the roots is equal to −c1 /c0 , the sum of the products of the roots taken two at a time is equal to c2 /c0 , the sum of the products of the roots taken three at a time is equal to −c3 /c0 , etc.; finally, the product of all the roots is equal to (−1)n cn /c0 Since we may . Ex. 2 the points representing the six sixth roots of unity. Obtain this result another way. 4. Find the five fifth roots of −1. 5. Obtain the trigonometric forms of the nine ninth roots of unity di- viding the members of the resulting equation by cos θ + i sin θ, we get cos β + i sin β cos θ + i sin θ = cos(β − θ) + i sin(β − θ). Hence the amplitude of the quotient of R(cos β+i sin β) by. The Project Gutenberg EBook of First Course in the Theory of Equations, by Leonard Eugene Dickson This eBook is for the use of anyone anywhere at no cost and